#107173

order	512
nilpotency class	3
isoclinism	isoclinic class of #107154
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,12,7,11,6,10)(3,13,15,8,17,24,16,23)(4,18,20,26,21,9,19,25)(14,28,30,22,31,27,29,32)(33,34)(35,43)(36,47,49,40)(37,50)(38,42)(39,45)(41,53)(44,59,61,64)(46,57)(48,56,52,58)(51,55)(54,60,63,62);(1,3,6,16,7,17,5,15)(2,8,12,24,11,23,10,13)(4,14,20,30,21,31,19,29)(9,22,26,28,18,32,25,27)(33,35)(34,39)(36,44)(37,51)(38,46)(40,54)(41,57)(42,55)(43,50)(45,53)(47,63)(48,62)(49,61)(52,60)(56,59)(58,64);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(23,32)(24,28)(33,36)(34,40)(35,44)(37,52)(38,49)(39,54)(41,58)(42,47)(43,59)(45,62)(46,61)(48,53)(50,56)(51,60)(55,63)(57,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ei,d^c=di,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14369);